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Journal articles on the topic 'Extended Variational Formulation (XVF)'

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Published: 4 June 2021

Last updated: 15 February 2022

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Blanco, Pablo, Paola Gervasio, and Alfio Quarteroni. "Extended Variational Formulation for Heterogeneous Partial Differential Equations." Computational Methods in Applied Mathematics 11, no. 2 (2011): 141–72. http://dx.doi.org/10.2478/cmam-2011-0008.

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AbstractWe address the coupling of an advection equation with a diffusion-advection equation, for solutions featuring boundary layers. We consider non-overlapping domain decompositions and we face up the heterogeneous problem using an extended variational formulation. We will prove the equivalence between the latter formulation and a treatment based on a singular perturbation theory. An exhaustive comparison in terms of solution and computational efficiency between these formulations is carried out.

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Noor, Muhammad Aslam. "Parametric Extended General Mixed Variational Inequalities." Journal of Applied Mathematics 2012 (2012): 1–11. http://dx.doi.org/10.1155/2012/201947.

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It is well known that the resolvent equations are equivalent to the extended general mixed variational inequalities. We use this alternative equivalent formulation to study the sensitivity of the extended general mixed variational inequalities without assuming the differentiability of the given data. Since the extended general mixed variational inequalities include extended general variational inequalities, quasi (mixed) variational inequalities and complementarity problems as special cases, results obtained in this paper continue to hold for these problems. In fact, our results can be considered as a significant extension of previously known results.

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Strozzi, Antonio. "The Elastohydrodynamic Problem Expressed in Terms of Extended Variational Formulation." Journal of Tribology 108, no. 4 (1986): 557–64. http://dx.doi.org/10.1115/1.3261263.

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The elastohydrodynamic problem is revisited in terms of an extended variational formulation, where the corresponding functional exhibits minimum properties in the solution neighborhood. Such features are exploited in the development of a relaxation-type solver. The numerical results indicate that the convergence rate of the proposed relaxation scheme becomes increasingly poor as the solution of the elastohydrodynamic problem is approached. A polyalgorithm based on a combination between relaxation-type and Newton-type schemes is proposed. The numerical experiments referred to various sealing profiles of increasing foundation compliance show that the proposed procedure is particularly advantageous in the case of soft lubricated contacts.

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Noor, Muhammad, Awais Khan, Khalida Noor, and Amjad Pervez. "Gauss-Seidel type algorithms for a class of variational inequalities." Filomat 32, no. 2 (2018): 395–407. http://dx.doi.org/10.2298/fil1802395n.

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In this paper, we consider a new system of extended general quasi variational inequalities involving six nonlinear operators. Using projection operator technique, we show that system of extended general quasi variational inequalities is equivalent to a system of fixed point problems. Using this alternative equivalent formulation, we propose and analyze Gauss-Seidel type algorithms for solving a system of extended general quasi variational inequalities. Convergence of new method is discussed under some suitable conditions. Several special cases are discussed. Results obtained in this paper continue to hold for these problems.

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Seshadri, R. "Limit Loads Using Extended Variational Concepts in Plasticity." Journal of Pressure Vessel Technology 122, no. 3 (2000): 379–85. http://dx.doi.org/10.1115/1.556196.

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Lower-bound limit load estimates are relevant from a standpoint of pressure component design, and are acceptable quantities for ascertaining primary stress limits. Elastic modulus adjustment procedures, used in conjunction with linear elastic finite element analyses, generate both statically admissible stress distributions and kinematically admissible strain distributions. Mura’s variational formulation for determining limit loads, originally developed as an alternative to the classical method, is extended further by allowing the elastic calculated stress fields to exceed yield provided they satisfy the “integral mean of yield” criterion. Consequently, improved lower-bound values for limit loads are obtained by solving a simple quadratic equation. The improved lower-bound limit load determination procedure, which is designated “the mα method,” is applied to symmetric as well as nonsymmetric components. [S0094-9930(00)01103-3]

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Ciaglia, F. M., F. Di Cosmo, A. Ibort, G. Marmo, and L. Schiavone. "Covariant Variational Evolution and Jacobi brackets: Fields." Modern Physics Letters A 35, no. 23 (2020): 2050206. http://dx.doi.org/10.1142/s0217732320502065.

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The analysis of the covariant brackets on the space of functions on the solutions to a variational problem in the framework of contact geometry initiated in the companion letter[Formula: see text] is extended to the case of the multisymplectic formulation of the free Klein–Gordon theory and of the free Schrödinger equation.

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WIO, HORACIO S. "VARIATIONAL FORMULATION FOR THE KPZ AND RELATED KINETIC EQUATIONS." International Journal of Bifurcation and Chaos 19, no. 08 (2009): 2813–21. http://dx.doi.org/10.1142/s0218127409024505.

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We present a variational formulation for the Kardar–Parisi–Zhang (KPZ) equation that leads to a thermodynamic-like potential for the KPZ as well as for other related kinetic equations. For the KPZ case, with the knowledge of such a potential we prove some global shift invariance properties previously conjectured by other authors. We also show a few results about the form of the stationary probability distribution function for arbitrary dimensions. The procedure used for KPZ was extended in order to derive more general forms of such a functional leading to other nonlinear kinetic equations, as well as cases with density dependent surface tension.

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Petrot, Narin, and Javad Balooee. "Algorithms for Solving System of Extended General Variational Inclusions and Fixed Points Problems." Abstract and Applied Analysis 2012 (2012): 1–27. http://dx.doi.org/10.1155/2012/569592.

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We introduce a new system of extended general nonlinear variational inclusions with different nonlinear operators and establish the equivalence between the aforesaid system and the fixed point problem. By using this equivalent formulation, we prove the existence and uniqueness theorem for solution of the system of extended general nonlinear variational inclusions. We suggest and analyze a new resolvent iterative algorithm to approximate the unique solution of the system of extended general nonlinear variational inclusions which is a fixed point of a nearly uniformly Lipschitzian mapping. Subsequently, the convergence analysis of the proposed iterative algorithm under some suitable conditions is considered. Furthermore, some related works to our main problem are pointed out and discussed.

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Balooee, Javad, Yeol Je Cho, and Mee Kwang Kang. "Projection Methods and a New System of Extended General Regularized Nonconvex Set-Valued Variational Inequalities." Journal of Applied Mathematics 2012 (2012): 1–18. http://dx.doi.org/10.1155/2012/690648.

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A new system of extended general nonlinear regularized nonconvex set-valued variational inequalities is introduced, and the equivalence between the extended general nonlinear regularized nonconvex set-valued variational inequalities and the fixed point problems is verified. Then, by this equivalent formulation, a new perturbed projection iterative algorithm with mixed errors for finding a solution of the aforementioned system is suggested and analyzed. Also the convergence of the suggested iterative algorithm under some suitable conditions is proved.

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Yasutake, Nobutoshi, and Shoichi Yamada. "Variational approach for rotating-stellar evolution in Lagrange scheme." Proceedings of the International Astronomical Union 9, S307 (2014): 150–51. http://dx.doi.org/10.1017/s1743921314006620.

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AbstractWe have developed an entirely new formulation to obtain self-gravitating, axisymmetric configurations in permanent rotation. It is based on the Lagrangian variational principle and, as a consequence, will allow us to apply it to stellar evolution calculations rather easily. We adopt a Monte Carlo technique, which is analogous to those employed in other fields, e.g. nuclear physics, in minimizing the energy functional. We also present the analogies between the study on rotating stellar configurations and the one on deformed nuclei. Possible applications are not limited to main sequence stars but will be extended to e.g. compact stars, proto-stars and planets. We believe that our formulation will be a major break-through then.

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de Angelis, Fabio, and Donato Cancellara. "Constitutive Equations for a Model of Nonlocal Plasticity which Complies with a Nonlocal Maximum Plastic Dissipation Principle." Applied Mechanics and Materials 217-219 (November 2012): 2362–66. http://dx.doi.org/10.4028/www.scientific.net/amm.217-219.2362.

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In the present paper constitutive equations for a nonlocal plasticity model are presented. Elasticity is considered to be governed by local forces so that only the dissipation processes are adopted as nonlocal. Differing from other proposed models in which the isotropic hardening/softening variables are considered as nonlocal, in the present paper the nonlocality is extended in order to include the kinematic hardening behaviour as well, so that both types of hardening (kinematic and isotropic) are considered as nonlocal. The present formulation satisfies a variational condition representing nonlocal maximum plastic dissipation. The proposed constitutive formulation of nonlocal plasticity is thus equipped with a sound variational basis.

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Urban, Zbyněk, and Ján Brajerčík. "The fundamental Lepage form in variational theory for submanifolds." International Journal of Geometric Methods in Modern Physics 15, no. 06 (2018): 1850103. http://dx.doi.org/10.1142/s0219887818501037.

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The multiple-integral variational functionals for finite-dimensional immersed submanifolds are studied by means of the fundamental Lepage equivalent of a homogeneous Lagrangian, which can be regarded as a generalization of the well-known Hilbert form in the classical mechanics. The notion of a Lepage form is extended to manifolds of regular velocities and plays a basic role in formulation of the variational theory for submanifolds. The theory is illustrated on the minimal submanifolds problem, including analysis of conservation law equations.

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Ting, Edward C., Chiang Shih, and Yeon-Kang Wang. "Fundamentals of a Vector Form Intrinsic Finite Element: Part II. Plane Solid Elements." Journal of Mechanics 20, no. 2 (2004): 123–32. http://dx.doi.org/10.1017/s1727719100003348.

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AbstractIn the second article of the series, the vector form intrinsic finite element is extended to formulate plane solid elements, a three-node triangular element and a four-node isoparametric element. Also, conceptual differences of the intrinsic element and traditional element based on variational formulation are discussed.

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Kijanski, Wojciech, and Franz-Joseph Barthold. "Two-scale shape optimisation based on numerical homogenisation techniques and variational sensitivity analysis." Computational Mechanics 67, no. 4 (2021): 1021–40. http://dx.doi.org/10.1007/s00466-020-01955-6.

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AbstractThis contribution presents a theoretical and computational framework for two-scale shape optimisation of nonlinear elastic structures. Particularly, minimum compliance optimisation problems with composite (matrix-inclusion) microstructures subjected to static loads and volume-type design constraints are focused. A homogenisation-based FE$$^2$$ 2 scheme is extended by an enhanced formulation of variational (shape) sensitivity analysis based on Noll’s intrinsic, frame-free formulation of continuum mechanics. The obtained overall two-scale sensitivity information couples shape variations across micro- and macroscopic scales. A numerical example demonstrates the capabilities of the proposed variational sensitivity analysis and the (shape) optimisation framework. The investigations involve a mesh morphing scheme for the design parametrisation at both macro- and microscopic scales.

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Huang, Xiang-Yu, Qingnong Xiao, Dale M. Barker, et al. "Four-Dimensional Variational Data Assimilation for WRF: Formulation and Preliminary Results." Monthly Weather Review 137, no. 1 (2009): 299–314. http://dx.doi.org/10.1175/2008mwr2577.1.

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Abstract The Weather Research and Forecasting (WRF) model–based variational data assimilation system (WRF-Var) has been extended from three- to four-dimensional variational data assimilation (WRF 4D-Var) to meet the increasing demand for improving initial model states in multiscale numerical simulations and forecasts. The initial goals of this development include operational applications and support to the research community. The formulation of WRF 4D-Var is described in this paper. WRF 4D-Var uses the WRF model as a constraint to impose a dynamic balance on the assimilation. It is shown to implicitly evolve the background error covariance and to produce the flow-dependent nature of the analysis increments. Preliminary results from real-data 4D-Var experiments in a quasi-operational setting are presented and the potential of WRF 4D-Var in research and operational applications are demonstrated. A wider distribution of the system to the research community will further develop its capabilities and to encourage testing under different weather conditions and model configurations.

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Zhang, Ping, and Xiaohua Zhang. "Numerical Modeling of Stokes Flow in a Circular Cavity by Variational Multiscale Element Free Galerkin Method." Mathematical Problems in Engineering 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/451546.

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The variational multiscale element free Galerkin method is extended to simulate the Stokes flow problems in a circular cavity as an irregular geometry. The method is combined with Hughes’s variational multiscale formulation and element free Galerkin method; thus it inherits the advantages of variational multiscale and meshless methods. Meanwhile, a simple technique is adopted to impose the essential boundary conditions which makes it easy to solve problems with complex area. Finally, two examples are solved and good results are obtained as compared with solutions of analytical and numerical methods, which demonstrates that the proposed method is an attractive approach for solving incompressible fluid flow problems in terms of accuracy and stability, even for complex irregular boundaries.

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Stango, R. J., and R. H. Jungmann. "A Variational Method for Evaluating Thrust Bearing Element Load Distribution." Journal of Engineering for Industry 111, no. 1 (1989): 79–86. http://dx.doi.org/10.1115/1.3188735.

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A variational method is outlined for computing thrust bearing element loads on the basis of minimizing the potential energy of the system. The problem is formulated in terms of a polynomial displacement assumption for bearing elements. To illustrate the computational procedure, numerical studies are presented for a thrust bearing subjected to a range of load eccentricities. The variational approach is demonstrated to result in an accurate and efficient solution for bearing element load distributions. Excellent agreement is achieved when comparison is made to conventional methods of classical bearing theory for nominal load eccentricities, while superior performance is obtained when load eccentricities are considerably larger. Basic advantages of the variational formulation are discussed and an illustrative problem is presented which demonstrates extended capability of the variational method for examining the load distribution in thrust bearings.

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Liu, H. Y., Na Si, and Ji-Huan He. "A short remark on Chien’s variational principle of maximum power losses for viscous fluids." International Journal of Numerical Methods for Heat & Fluid Flow 26, no. 3/4 (2016): 694–97. http://dx.doi.org/10.1108/hff-09-2015-0368.

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Purpose – The purpose of this paper is to point out a paradox in variational theory for viscous flows. Chien (1984) claimed that a variational principle of maximum power loses for viscous fluids was established, however, it violated the well-known Helmholtz’s principle. Design/methodology/approach – Restricted variables are introduced in the derivation, the first order and the second order of variation of the restricted variables are zero. Findings – An approximate variational principle of minimum power loses is established, which agrees with the Helmholtz’s principle, and the paradox is solved. Research limitations/implications – This paper focusses on incompressible viscose flows, and the theory can be extended to compressible one and other viscose flows. It is still difficult to obtain a variational formulation for Navier-Stokes equations. Practical implications – The variational principle of minimum power loses can be directly used for numerical methods and analytical analysis. Originality/value – It is proved that Chien’s variational principle is a minimum principle.

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Zhavoronok, Sergey I. "On the Variational Formulation of the Extended Thick Anisotropic Shells Theory of I. N. Vekua Type." Procedia Engineering 111 (2015): 888–95. http://dx.doi.org/10.1016/j.proeng.2015.07.164.

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Svendsen, Bob, and Swantje Bargmann. "On the continuum thermodynamic rate variational formulation of models for extended crystal plasticity at large deformation." Journal of the Mechanics and Physics of Solids 58, no. 9 (2010): 1253–71. http://dx.doi.org/10.1016/j.jmps.2010.06.005.

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Lebon, G., and P. C. Dauby. "Heat transport in dielectric crystals at low temperature: A variational formulation based on extended irreversible thermodynamics." Physical Review A 42, no. 8 (1990): 4710–15. http://dx.doi.org/10.1103/physreva.42.4710.

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CARTER, BRANDON, and NICOLAS CHAMEL. "COVARIANT ANALYSIS OF NEWTONIAN MULTI-FLUID MODELS FOR NEUTRON STARS I: MILNE–CARTAN STRUCTURE AND VARIATIONAL FORMULATION." International Journal of Modern Physics D 13, no. 02 (2004): 291–325. http://dx.doi.org/10.1142/s0218271804004542.

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This is the first of a series of articles showing how 4 dimensionally covariant analytical procedures developed in the context of General Relativity can be usefully adapted for application in a purely Newtonian framework where they provide physical insights (e.g. concerning helicity currents) that are not so easy to obtain by the traditional approach based on a 3+1 spacetime decomposition. After an introductory presentation of the relevant Milne spacetime structure and the associated Cartan connection, the essential principles are illustrated by application to the variational formulation of simple barotropic perfect fluid models. This variational treatment is then extended to conservative multiconstituent self-gravitating fluid models of the more general kind that is needed for treating the effects of superfluidity in neutron stars.

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Kim, Jinkyu, and Dongkeon Kim. "Temporal finite element methods through the extended framework of Hamilton’s principle." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 231, no. 2 (2016): 263–78. http://dx.doi.org/10.1177/0954406216642481.

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With basic ideas of mixed Lagrangian formulation and sequential assigning process for initial conditions, the extended framework of Hamilton’s principle (EHP) was recently developed for continuum dynamics. Unlike the original Hamilton’s principle, this new variational framework can fully take initial conditions into account for both linear and nonlinear dynamics, so that it provides a sound base to apply a finite element scheme over the temporal domain without any ambiguity. This paper describes temporal finite element approach stemming from the extended Hamilton’s principle, which focuses initially on classical single-degree-of-freedom oscillators such as Kelvin–Voigt damped oscillator and an elasto-viscoplastic model. In each case, an appropriate weak form is provided and a corresponding formulation is discretized in the temporal domain with the adoption of Galerkin’s method. Basic numerical properties are investigated for the developed numerical algorithms with several computational examples for the elasto-viscoplastic model. For the underlying conservative system, the present method is symplectic and unconditionally stable with respect to the time step. On the other hand, the method provides unconditionally stable and noniterative algorithm for the elasto-viscoplastic model.

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Guermond, J. L., and L. Quartapelle. "On Sensitive Vector Poisson and Stokes Problems." Mathematical Models and Methods in Applied Sciences 07, no. 05 (1997): 681–98. http://dx.doi.org/10.1142/s0218202597000360.

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Lions/Sanchez-Palencia's theory of sensitive boundary value problems is extended from the scalar biharmonic equation to the vector Poisson equation and the Stokes problem associated with the bilinear form (∇ × u, ∇ × v) + (∇ · u, ∇ · v). For both problems the specification of completely natural conditions for the vector unknown on a part of the boundary leads to a variational formulation admitting a unique solution which is however sensitive to abitrarily small smooth perturbations of the data, as shown in the present paper.

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Miehe, C., S. Teichtmeister, and F. Aldakheel. "Phase-field modelling of ductile fracture: a variational gradient-extended plasticity-damage theory and its micromorphic regularization." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 374, no. 2066 (2016): 20150170. http://dx.doi.org/10.1098/rsta.2015.0170.

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This work outlines a novel variational-based theory for the phase-field modelling of ductile fracture in elastic–plastic solids undergoing large strains. The phase-field approach regularizes sharp crack surfaces within a pure continuum setting by a specific gradient damage modelling. It is linked to a formulation of gradient plasticity at finite strains. The framework includes two independent length scales which regularize both the plastic response as well as the crack discontinuities. This ensures that the damage zones of ductile fracture are inside of plastic zones, and guarantees on the computational side a mesh objectivity in post-critical ranges.

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Daneshmehr, Ali Reza, Majid Akbarzadeh Khorshidi, and Delara Soltani. "Dynamic Analysis of a Micro-Cantilever Subjected to Harmonic Base Excitation via RVIM." Applied Mechanics and Materials 332 (July 2013): 545–50. http://dx.doi.org/10.4028/www.scientific.net/amm.332.545.

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In this paper, dynamic analysis of a cantilever beam with micro-scale dimensions is presented. The micro-cantilever is subjected to harmonic base excitation and constant force at micro-cantilever tip. By Euler-Bernoulli beam theory assumptions, the mathematical formulation of vibrating micro-cantilever beam is derived using extended Hamilton principle. The governing partial-diffrential equation is solved by reconstruction of variational iteration method (RVIM), with possession of its boundary conditions. The RVIM is an approximate method of solving that answers easy and quick and has high accuracy.

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Qian, Duan, and J. S. Hansen. "A Time Domain Substructure Synthesis Method for Viscoelastic Structures." Journal of Applied Mechanics 62, no. 2 (1995): 407–13. http://dx.doi.org/10.1115/1.2895945.

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The method of substructure synthesis, originally conceived for undamped and viscously damped systems, has been extended to systems with viscoelastic damping in the hereditary integral form. Based on a new variational principle, the substructure synthesis method is formulated in the time domain. The displacement in each substructure is represented by a set of real admissible trial vectors. The traditional state space formulation is avoided by the proposed method so that the approach is independent of the form of viscoelastic models. Effectiveness of the method is illustrated through numerical examples.

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Hackl, Klaus, and Franz Dieter Fischer. "On the relation between the principle of maximum dissipation and inelastic evolution given by dissipation potentials." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 464, no. 2089 (2007): 117–32. http://dx.doi.org/10.1098/rspa.2007.0086.

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We study the evolution of systems described by internal variables. After the introduction of thermodynamic forces and fluxes, both the dissipation and dissipation potential are defined. Then, the principle of maximum dissipation (PMD) and a minimum principle for the dissipation potential are developed in a variational formulation. Both principles are related to each other. Several cases are shown where both principles lead to the same evolution equations for the internal variables. However, also counterexamples are reported where such an equivalence is not valid. In this case, an extended PMD can be formulated.

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Shen, I. Y. "A Variational Formulation, a Work-Energy Relation and Damping Mechanisms of Active Constrained Layer Treatments." Journal of Vibration and Acoustics 119, no. 2 (1997): 192–99. http://dx.doi.org/10.1115/1.2889702.

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The purposes of this paper are to formulate active constrained layer (ACL) damping treatments through a variational approach, to study the work-energy relation of ACL, and to identify damping mechanisms of ACL treatments. Application of the extended Hamilton principle to ACL results in the equations of motion of ACL and the charge equation of electrostatics for the piezoelectric constraining layer. The work-energy equation together with the charge equation shows that the power dissipated through the active damping is the product of the electric field and the axial velocity of the piezoelectric constraining layer at the boundaries. This unique feature suggests that a self-sensing and actuating piezoelectric constraining layer may be an appropriate design in dissipating vibration energy without causing instability. To identify the damping mechanisms, a sensitivity analysis shows that the effectiveness of ACL damping primarily depends on the active and passive damping forces transmitted to the vibrating structure through the viscoelastic layer. The active damping force transmitted depends on the controller transfer function as well as a system parameter, termed active damping sensitivity factor, which depends entirely on the configuration of the passive constrained layer and the sensor. Finally, numerical results on ACL beams are obtained to illustrate the theoretical predictions above.

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Gwinner, J. "Three-Field Modelling of Nonlinear Nonsmooth Boundary Value Problems and Stability of Differential Mixed Variational Inequalities." Abstract and Applied Analysis 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/108043.

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The purpose of this paper is twofold. Firstly we consider nonlinear nonsmooth elliptic boundary value problems, and also related parabolic initial boundary value problems that model in a simplified way steady-state unilateral contact with Tresca friction in solid mechanics, respectively, stem from nonlinear transient heat conduction with unilateral boundary conditions. Here a recent duality approach, that augments the classical Babuška-Brezzi saddle point formulation for mixed variational problems to twofold saddle point formulations, is extended to the nonsmooth problems under consideration. This approach leads to variational inequalities of mixed form for three coupled fields as unknowns and to related differential mixed variational inequalities in the time-dependent case. Secondly we are concerned with the stability of the solution set of a general class of differential mixed variational inequalities. Here we present a novel upper set convergence result with respect to perturbations in the data, including perturbations of the associated nonlinear maps, the nonsmooth convex functionals, and the convex constraint set. We employ epiconvergence for the convergence of the functionals and Mosco convergence for set convergence. We impose weak convergence assumptions on the perturbed maps using the monotonicity method of Browder and Minty.

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Atalla, N., and J. Nicolas. "A Formulation for Mean Flow Effects on Sound Radiation from Rectangular Baffled Plates with Arbitrary Boundary Conditions." Journal of Vibration and Acoustics 117, no. 1 (1995): 22–29. http://dx.doi.org/10.1115/1.2873863.

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A general formulation of the sound radiation from fluid-loaded rectangular baffled plates with arbitrary boundary conditions has been developed by Berry et al. (JASA, Vol. 90, No. 4, Pt. 2, 1991). In this paper, an extension of this formulation to inviscid, uniform subsonic flow is considered. The analysis is based on a variational formulation for the transverse vibrations of the plate and the use of the extended, to uniformly moving media, form of the Helmholtz integral equation. The formulation shows explicitly the effect of the flow in terms of added mass, and radiation resistance. Furthermore, it avoids the difficult problem of integration in the complex domain, typical of the wavenumber transform approaches to fluid-loading problems. Comparison of the acoustic radiation impedance with existing studies supports the validity of the approach. The details of the formulation and its numerical implementation is exposed and a discussion of the flow effects on the radiation impedance of a rectangular piston is presented. It is shown that subsonic mean flow increases the modal radiation resistance at low frequencies and affects added mass more strongly than it affects radiation resistance.

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Sato, M., Y. Konishi, and S. J. Park. "Interlayer Coupling Effect on Buckling Modes of Spherical Bilayers." Journal of Mechanics 31, no. 1 (2014): 29–36. http://dx.doi.org/10.1017/jmech.2014.64.

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AbstractThis study examined the critical buckling characteristics of hydrostatically pressurized double-walled complete spherical shells. An analytical model based on small deflection thin shell theory is presented; the equations are solved in conjunction with variational principles. Axisymmetric and inextensional assumptions are not initially used in the exact formulation. This approach therefore avoids any discussion about the validity of the solution and allows the model to be extended to cover more generic nonaxisymmetric cases with relative ease. The analytical results are presented using illustrative buckling modes. Based on the developed formulation, only axisymmetric eigenmodes were found to occur despite the inclusion of the effect of interactions between outer and inner shells. Critical modes that are symmetric or antisymmetric about the equator may be determined depending on the combination of the stiffness connecting the outer and inner shells and the radius-to-wall thickness ratios.

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Pan, L., and R. Seshadri. "Limit Load Estimation Using Plastic Flow Parameter in Repeated Elastic Finite Element Analyses." Journal of Pressure Vessel Technology 124, no. 4 (2002): 433–39. http://dx.doi.org/10.1115/1.1499960.

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The procedures described in this paper for determining a limit load is based on Mura’s extended variational formulation. Used in conjunction with linear elastic finite element analyses, the approach provides a robust method to estimate limit loads of mechanical components and structures. The secant modulus of the various elements in a finite element discretization scheme is prescribed in order to simulate the distributed effect of the plastic flow parameter, μ0. The upper and lower-bound multipliers m0 and m′ obtained using this formulation converge to near exact values. By using the notion of “leap-frogging” to limit state, an improved lower-bound multiplier, mα, can be obtained. The condition for which mα is a reasonable lower bound is discussed in this paper. The method is applied to component configurations such as cylinder, torispherical head, indeterminate beam, and a cracked specimen.

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Kulikov, GM, SV Plotnikova, and E. Carrera. "Modeling and analysis of spiral actuators by exact geometry piezoelectric solid-shell elements." Journal of Intelligent Material Systems and Structures 31, no. 1 (2019): 53–70. http://dx.doi.org/10.1177/1045389x19880014.

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An exact geometry four-node piezoelectric solid-shell element through the sampling surfaces formulation is proposed. The sampling surfaces formulation is based on choosing inside the shell N – 2 sampling surfaces parallel to the middle surface and located at Chebyshev polynomial nodes to introduce the displacements and electric potentials of these surfaces as fundamental shell unknowns. The bottom and top surfaces are also included into a set of sampling surfaces. Such choice of unknowns with the use of Lagrange polynomials of degree N – 1 in the through-the-thickness interpolations of displacements, strains, electric potential, and electric field yields a robust piezoelectric shell formulation. To implement efficient analytical integration throughout the solid-shell element, the extended assumed natural strain method is employed. The developed hybrid-mixed four-node piezoelectric solid-shell element is based on the Hu-Washizu variational principle and shows the excellent performance for coarse mesh configurations. It can be useful for the 3D stress analysis of piezoelectric shells with variable curvatures, in particular for the modeling and analysis of spiral actuators.

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Li, ShuGuang. "A cell-based multiple vehicle type dynamic user equilibrium model with physical queues." Canadian Journal of Civil Engineering 43, no. 1 (2016): 1–12. http://dx.doi.org/10.1139/cjce-2014-0231.

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This paper proposes a cell-based multiple vehicle type dynamic user equilibrium model with physical queues. A single-type traffic flow model is extended to a general case with multiple vehicle types that can be partly solved by the time-space discretization method. Then, a network version of the multiple vehicle type cell transmission model is given. An integrated variational inequality (VI) formulation is presented to capture the complex traveler choice behaviors such as route and departure time choices. Furthermore, a genetic algorithm with a flow-swapping method is adopted to solve the VI problem. Two examples are used to evaluate the properties of this formulation. The results show that the model can reflect dynamic phenomena, such as multiple vehicle type speed consistent under congested conditions, queue formation and dissipation and so on. Moreover, the solutions can approximately follow the multiple vehicle type dynamic route and departure time user equilibrium conditions.

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Zhang, Liang, Huiting Zhang, Jian Wu, Bo Yan, and Mengkai Lu. "Parametric Variational Principle for Bi-Modulus Materials and Its Application to Nacreous Bio-Composites." International Journal of Applied Mechanics 08, no. 06 (2016): 1650082. http://dx.doi.org/10.1142/s1758825116500824.

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Bi-modulus materials have different moduli in tension and compression and the stress–strain relation depends on principal stress that is unknown before displacement is determined. Establishment of variational principle is important for mechanical analysis of materials. First, parametric variational principle (PVP) is proposed for static analysis of bi-modulus materials and structures. A parametric variable indicating state of principal stress is included in the potential energy formulation and the nonlinear stress–strain relation is evolved into a linear complementarity constraint. Convergence of finite element analysis is thus improved. Then the proposed variational principle is extended to a dynamic problem and the dynamic equation can be derived based on Hamilton’s principle. Finite element analysis of nacreous bio-composites is performed, in which a unilateral contact behavior between two hard mineral bricks is modeled using the bi-modulus stress–strain relation. Effective modulus of composites can be determined numerically and stress mechanism of “tension–shear chain” in nacre is revealed. A delayed effect on stress propagation is found around the “gaps” between mineral bricks, when a tension force is loaded to nacreous bio-composites dynamically.

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Dumont, N. A. "The Hybrid Boundary Element Method: An Alliance Between Mechanical Consistency and Simplicity." Applied Mechanics Reviews 42, no. 11S (1989): S54—S63. http://dx.doi.org/10.1115/1.3152408.

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The most important features of a new numerical method are outlined. The mechanical, or variational consistency of the hybrid finite element method is extended to the conventional boundary element formulation, giving rise to naturally established symmetric force-displacement relations. The computational effort for the complete solution of a given problem, according to this method, is in some cases only a small fraction of the effort needed with traditional methods. This paper also outlines briefly the types of analyses which may be advantageously performed with this new method, many of which are already being implemented by the author and co-workers. Some numerical examples are provided.

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Li, Shaofan, Anurag Gupta, and Xanthippi Markenscoff. "Conservation laws of linear elasticity in stress formulations." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 461, no. 2053 (2005): 99–116. http://dx.doi.org/10.1098/rspa.2004.1347.

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In this paper, we present new conservation laws of linear elasticity which have been discovered. These newly discovered conservation laws are expressed solely in terms of the Cauchy stress tensor, and they are genuine, non–trivial conservation laws that are intrinsically different from the displacement conservation laws previously known. They represent the variational symmetry conditions of combined Beltrami–Michell compatibility equations and the equilibrium equations. To derive these conservation laws, Noether's theorem is extended to partial differential equations of a tensorial field with general boundary conditions. By applying the tensorial version of Noether's theorem to Pobedrja's stress formulation of three–dimensional elasticity, a class of new conservation laws in terms of stresses has been obtained.

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Liu, Chengsi, and Ming Xue. "Relationships among Four-Dimensional Hybrid Ensemble–Variational Data Assimilation Algorithms with Full and Approximate Ensemble Covariance Localization." Monthly Weather Review 144, no. 2 (2016): 591–606. http://dx.doi.org/10.1175/mwr-d-15-0203.1.

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Abstract Ensemble–variational data assimilation algorithms that can incorporate the time dimension (four-dimensional or 4D) and combine static and ensemble-derived background error covariances (hybrid) are formulated in general forms based on the extended control variable and the observation-space-perturbation approaches. The properties and relationships of these algorithms and their approximated formulations are discussed. The main algorithms discussed include the following: 1) the standard ensemble 4DVar (En4DVar) algorithm incorporating ensemble-derived background error covariance through the extended control variable approach, 2) the 4DEnVar neglecting the time propagation of the extended control variable (4DEnVar-NPC), 3) the 4D ensemble–variational algorithm based on observation space perturbation (4DEnVar), and 4) the 4DEnVar with no propagation of covariance localization (4DEnVar-NPL). Without the static background error covariance term, none of the algorithms requires the adjoint model except for En4DVar. Costly applications of the tangent linear model to localized ensemble perturbations can be avoided by making the NPC and NPL approximations. It is proven that En4DVar and 4DEnVar are mathematically equivalent, while 4DEnVar-NPC and 4DEnVar-NPL are mathematically equivalent. Such equivalences are also demonstrated by single-observation assimilation experiments with a 1D linear advection model. The effects of the non-flow-following or stationary localization approximations are also examined through the experiments. All of the above algorithms can include the static background error covariance term to establish a hybrid formulation. When the static term is included, all algorithms will require a tangent linear model and an adjoint model. The first guess at appropriate time (FGAT) approximation is proposed to avoid the tangent linear and adjoint models. Computational costs of the algorithms are also discussed.

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Civelek, Cem. "Analysis of a coupled physical discrete time system by means of extended Euler-Lagrange difference equation and discrete dissipative canonical equation." COMPEL - The international journal for computation and mathematics in electrical and electronic engineering 38, no. 6 (2019): 1810–27. http://dx.doi.org/10.1108/compel-04-2019-0163.

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Purpose The purpose of this study is the application of the following concepts to the time discrete form. Variational Calculus, potential and kinetic energies, velocity proportional Rayleigh dissipation function, the Lagrange and Hamilton formalisms, extended Hamiltonians and Poisson brackets are all defined and applied for time-continuous physical processes. Such processes are not always time-continuously observable; they are also sometimes time-discrete. Design/methodology/approach The classical approach is developed with the benefit of giving only a short table on charge and flux formulation, as they are similar to the classical case just like all other formulation types. Moreover, an electromechanical example is represented as well. Findings Lagrange and Hamilton formalisms together with the velocity proportional (Rayleigh) dissipation function can also be used in the discrete time case, and as a result, dissipative equations of generalized motion and dissipative canonical equations in the discrete time case are obtained. The discrete formalisms are optimal approaches especially to analyze a coupled physical system which cannot be observed continuously. In addition, the method makes it unnecessary to convert the quantities to the other. The numerical solutions of equations of dissipative generalized motion of an electromechanical (coupled) system in continuous and discrete time cases are presented. Originality/value The formalisms and the velocity proportional (Rayleigh) dissipation function aforementioned are used and applied to a coupled physical system in time-discrete case for the first time to the best of the author’s knowledge, and systems of difference equations are obtained depending on formulation type.

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Tu, Shuangzhang, Gordon W. Skelton, and Qing Pang. "Extension Of The High-Order Space-Time Discontinuous Galerkin Cell Vertex Scheme To Solve Time Dependent Diffusion Equations." Communications in Computational Physics 11, no. 5 (2012): 1503–24. http://dx.doi.org/10.4208/cicp.050810.090611a.

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AbstractIn this paper, the high-order space-time discontinuous Galerkin cell vertex scheme (DG-CVS) developed by the authors for hyperbolic conservation laws is extended for time dependent diffusion equations. In the extension, the treatment of the diffusive flux is exactly the same as that for the advective flux. Thanks to the Riemann-solver-free and reconstruction-free features of DG-CVS, both the advective flux and the diffusive flux are evaluated using continuous information across the cell interface. As a result, the resulting formulation with diffusive fluxes present is still consistent and does not need any extra ad hoc techniques to cure the common “variational crime” problem when traditional DG methods are applied to diffusion problems. For this reason, DG-CVS is conceptually simpler than other existing DG-typed methods.

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Ali, Haider, Amna Shujjahuddin, and Lavdie Rada. "A New Active Contours Image Segmentation Model Driven by Generalized Mean with Outlier Restoration Achievements." International Journal of Pattern Recognition and Artificial Intelligence 34, no. 11 (2020): 2054026. http://dx.doi.org/10.1142/s0218001420540269.

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In this paper, we propose a robust variational segmentation model capable of overcoming the problem of the negative effects of outliers. The proposed method is based on the combination of the characteristics of the generalized mean with the concept of the active contours. The optimization problem raising in this combination employs a power method technique. We demonstrate the performance of the proposed model on a series of sample images from diverse modalities and show an outperforming proposed model in comparison with the state-of-the-art methods. The proposed method shows better or equivalent performance in terms of accuracy and robustness than the conventional state-of-the-art models. The validation of the efficiency of the proposed two-phase algorithm is further extended to vector-valued images and multi-phase formulation.

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Li, Xiao Chuan, and Jin Shuang Zhang. "Mixed State Hamiltonian Element for Plane Transversely Isotropic Magnetoelectroelastic Solids." Applied Mechanics and Materials 275-277 (January 2013): 1978–83. http://dx.doi.org/10.4028/www.scientific.net/amm.275-277.1978.

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Hamiltonian dual equation of plane transversely isotropic magnetoelectroelastic solids is derived from variational principle and mixed state Hamiltonian elementary equations are established. Similar to the Hamiltonian formulation in classic dynamics, the z coordinate is treated analogous to the time coordinate. Then the x-direction is discreted with the linear elements to obtain the state-vector governing equations, which are a set of first order differential equations in z and are solved by the analytical approach. Because present approach is analytic in z direction, there is no restriction on the thickness of plate through the use of the present element. Using the propagation matrix method, the approach can be extended to analyze the problems of magnetoelectroelastic laminated plates. Present semi-analytical method of mixed Hamiltonian element has wide application area.

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Kulikov, G. M., and S. V. Plotnikova. "Exact geometry SaS solid-shell element for 3D stress analysis of FGM piezoelectric structures." Curved and Layered Structures 5, no. 1 (2018): 116–35. http://dx.doi.org/10.1515/cls-2018-0009.

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Abstract A hybrid-mixed functionally graded material (FGM) piezoelectric four-node solid-shell element through the sampling surfaces (SaS) method is proposed. The SaS formulation is based on choosing inside the shell N SaS parallel to the middle surface in order to introduce the displacements and electric potentials of these surfaces as fundamental shell unknowns. Such choice of unknowns with the use of Lagrange polynomials of degree N-1 in through-thickness interpolations of the displacements, strains, electric potential, electric field and material properties leads to a robust FGM piezoelectric shell formulation. The inner SaS are located at Chebyshev polynomial nodes that make it possible to minimize uniformly the error due to Lagrange interpolation. To implement the effective analytical integration throughout the element, the extended assumed natural strain (ANS) method is employed. As a result, the piezoelectric four-node solid-shell element exhibits a superior performance in the case of coarse meshes. To circumvent shear and membrane locking, the hybrid stress-strain solid-shell formulation via the Hu-Washizu variational principle is employed. The developed solid-shell element could be useful for the 3D stress analysis of FGMstructures because the SaS method allows obtaining the solutions with a prescribed accuracy, which asymptotically approach the exact solutions of electroelasticity as the number of SaS tends to infinity.

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KERSWELL, R. R. "Upper bounds on general dissipation functionals in turbulent shear flows: revisiting the ‘efficiency’ functional." Journal of Fluid Mechanics 461 (June 25, 2002): 239–75. http://dx.doi.org/10.1017/s0022112002008303.

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We show how the variational formulation introduced by Doering & Constantin to rigorously bound the long-time-averaged total dissipation rate [ ] in turbulent shear flows can be extended to treat other long-time-averaged functionals lim supT→∞(1/T)×∫0Tf(D, Dm, Dv)dt of the total dissipation D, dissipation in the mean field Dm and dissipation in the fluctuation field Dv. Attention is focused upon the suite of functionals f = D(Dv/Dm)n and f = Dm(Dv/Dm)n (n [ges ] 0) which include the ‘efficiency’ functional f = D(Dv/Dm) (Malkus & Smith 1989; Smith 1991) and the dissipation in the mean flow f = Dm (Malkus 1996) as special cases. Complementary lower estimates of the rigorous bounds are produced by generalizing Busse's multiple-boundary-layer trial function technique to the appropriate Howard–Busse variational problems built upon the usual assumption of statistical stationarity and constraints of total power balance, mean momentum balance, incompressibility and boundary conditions. The velocity field that optimizes the ‘efficiency’ functional is found not to capture the asymptotic structure of the observed mean flow in either plane Couette flow or plane Poiseuille flow. However, there is evidence to suppose that it is ‘close’ to a neighbouring functional that may.

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ANDREU, FUENSANTA, JOSÉ M. MAZÓN, and MIRCEA SOFONEA. "ENTROPY SOLUTIONS IN THE STUDY OF ANTIPLANE SHEAR DEFORMATIONS FOR ELASTIC SOLIDS." Mathematical Models and Methods in Applied Sciences 10, no. 01 (2000): 99–126. http://dx.doi.org/10.1142/s0218202500000082.

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The concept of entropy solution was recently introduced in the study of Dirichlet problems for elliptic equations and extended for parabolic equations with nonlinear boundary conditions. The aim of this paper is to use the method of entropy solutions in the study of a new problem which arise in the theory of elasticity. More precisely, we consider here the infinitesimal antiplane shear deformation of a cylindrical elastic body subjected to given forces and in a frictional contact with a rigid foundation. The elastic constitutive law is physically nonlinear and the friction is described by a static law. We present a variational formulation of the model and prove the existence and the uniqueness of a weak solution in the case when the body forces and the prescribed surface tractions have the regularity L∞. The proof is based on classical results for elliptic variational inequalities and measure theory arguments. We also define the concept of entropy solution and we prove an existence and uniqueness result in the case when the body forces and the surface tractions have the regularity L1. The proof is based on properties of the trace operators for functions which are not in Sobolev spaces. Finally, we present a regularity result for the entropy solution and we give some concrete examples and mechanical interpretation.

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Bridges, Thomas J. "Bifurcation of periodic solutions near a collision of eigenvalues of opposite signature." Mathematical Proceedings of the Cambridge Philosophical Society 108, no. 3 (1990): 575–601. http://dx.doi.org/10.1017/s0305004100069462.

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AbstractWhen two purely imaginary eigenvalues of opposite Krein signature coalesce, in a Hamiltonian system, a small perturbation can drive them off of the imaginary axis resulting in a linear instability. The most celebrated example of this instability occurs in the restricted 3-body problem at Routh's critical mass ratio. In this paper the collision of eigenvalues is treated as a singularity. A variational form of the Lyapunov–Schmidt method and distinguished parameter ࡃ2-equivariant singularity theory, with the frequency as distinguished parameter, are used to determine the effect of the degeneracy on the branches of periodic solutions in a neighbourhood. Previous results of Meyer and Schmidt[13], Sokol'skij [16] and van der Meer [12] are recovered in the formulation as a co-dimension 1 singularity. The results are extended to include the effect of an additional degeneracy (a co-dimension 2 singularity). The theory is applied to a spinning double pendulum.

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Neff, Patrizio. "Existence of minimizers for a finite-strain micromorphic elastic solid." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 136, no. 5 (2006): 997–1012. http://dx.doi.org/10.1017/s0308210500004844.

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We investigate geometrically exact generalized continua of micromorphic type in the sense of Eringen. The two-field problem for the macrodeformation φ and the affine microdeformation P̄ ∈ GL+(3, R) in the quasistatic, conservative load case is investigated in a variational form. Depending on material constants, two existence theorems in Sobolev spaces are given for the resulting nonlinear boundary-value problems. These results comprise existence results for the micro-incompressible case P̄ ∈ SL(3, R) and the Cosserat micropolar case P̄ ∈ SO(3, R). In order to treat external loads, a new condition, called bounded external work, has to be included, which overcomes the conditional coercivity of the formulation. The possible lack of coercivity is related to fracture of the micromorphic solid. The mathematical analysis uses an extended Korn first inequality. The methods of choice are the direct methods of the calculus of variations.

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Gabrielli, Davide, and D. R. Michiel Renger. "Dynamical Phase Transitions for Flows on Finite Graphs." Journal of Statistical Physics 181, no. 6 (2020): 2353–71. http://dx.doi.org/10.1007/s10955-020-02667-0.

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AbstractWe study the time-averaged flow in a model of particles that randomly hop on a finite directed graph. In the limit as the number of particles and the time window go to infinity but the graph remains finite, the large-deviation rate functional of the average flow is given by a variational formulation involving paths of the density and flow. We give sufficient conditions under which the large deviations of a given time averaged flow is determined by paths that are constant in time. We then consider a class of models on a discrete ring for which it is possible to show that a better strategy is obtained producing a time-dependent path. This phenomenon, called a dynamical phase transition, is known to occur for some particle systems in the hydrodynamic scaling limit, which is thus extended to the setting of a finite graph.

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Barfoot, Timothy D., James R. Forbes, and David J. Yoon. "Exactly sparse Gaussian variational inference with application to derivative-free batch nonlinear state estimation." International Journal of Robotics Research 39, no. 13 (2020): 1473–502. http://dx.doi.org/10.1177/0278364920937608.

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We present a Gaussian variational inference (GVI) technique that can be applied to large-scale nonlinear batch state estimation problems. The main contribution is to show how to fit both the mean and (inverse) covariance of a Gaussian to the posterior efficiently, by exploiting factorization of the joint likelihood of the state and data, as is common in practical problems. This is different than maximum a posteriori (MAP) estimation, which seeks the point estimate for the state that maximizes the posterior (i.e., the mode). The proposed exactly sparse Gaussian variational inference (ESGVI) technique stores the inverse covariance matrix, which is typically very sparse (e.g., block-tridiagonal for classic state estimation). We show that the only blocks of the (dense) covariance matrix that are required during the calculations correspond to the non-zero blocks of the inverse covariance matrix, and further show how to calculate these blocks efficiently in the general GVI problem. ESGVI operates iteratively, and while we can use analytical derivatives at each iteration, Gaussian cubature can be substituted, thereby producing an efficient derivative-free batch formulation. ESGVI simplifies to precisely the Rauch–Tung–Striebel (RTS) smoother in the batch linear estimation case, but goes beyond the ‘extended’ RTS smoother in the nonlinear case because it finds the best-fit Gaussian (mean and covariance), not the MAP point estimate. We demonstrate the technique on controlled simulation problems and a batch nonlinear simultaneous localization and mapping problem with an experimental dataset.

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